In this paper, as a continuation of [Fernandez-Guasti, \textit{ Celest Mech Dyn Astron} 137, 4 (2025)], we demonstrate the maximal superintegrability of the reduced Hamiltonian, which governs the four-body choreographic planar motion along the trisectrix lima\c{ c}on (resembling a folded figure eight), in the six-dimensional space of relative motion. The pairwise interaction potential $V(r_{ ij})$ among the four bodies is a quadratic expression in the relative distances $r_{ ij}$, with a combination of positive and negative coefficients. The corresponding eleven integrals of motion in the Liouville-Arnold sense are presented explicitly. Specifically, it is shown that the reduced Hamiltonian admits complete separation of variables in Jacobi-like variables. The emergence of this choreography is not a direct consequence of maximal superintegrability. Rather, it originates from the existence of \textit{ particular integrals} and the phenomenon of \textit{ particular involution}. We also provide a detailed analysis of the fragmentation of a general four-body choreographic motion into two isomorphic two-body choreographies, as well as the reverse process, namely, the fusion of two-body choreographies into a four-body configuration. This model combines choreographic motion with maximal superintegrability, a seldom-studied interplay in classical mechanics.
Categories: math-ph math.MP